# I³MS - Lockerby Seminar

## Prof. Duncan Lockerby, Ph.D. - Simulating Low-Speed Micro-Scale Gas Flows: A Method of Fundamental Solutions

School of Engineering, University of Warwick, UK

### Abstract

Creeping flow (also known as Stokes flow) is flow at vanishingly small Reynolds number. This classic topic has many applications, particularly in micro- and nano-flow technology, owing to both the small scale and the low speeds encountered. In these conditions the inertia of the fluid can be neglected, and (when local-equilibirum assumptions are valid) the momentum and continuity equation reduce to the Stokes equations. A common mathematical tool for the analysis of creeping flows is a fundamental singularity solution to the Stokes equations, known as the *Stokeslet* (first derived by Lorentz in 1897). This fundamental solution – in essence a Green’s function for the Stokes equations – is the flow response to a Dirac delta forcing term applied to the momentum equation.

In the most straightforward use, a flow field is represented by a superposition of Stokeslets (positioned outside of the domain) that are given a combination of strengths chosen to satisfy the same number of conditions at nodes on the boundary. This approach is known as the method of fundamental solutions (MFS) or the superposition method. The approach has the advantage of having a flow domain that is meshless, and a dimensionality that is reduced in order by one (the boundary is discretized rather than the volume). The Stokeslet is also in itself of interest, as a fundamental solution to the Stokes equations, and can be used to conveniently derive certain analytical results.

In dilute gas flows departed far from local thermodynamic equilibrium (i.e. at non negligible Knudsen numbers, such as in rarefied conditions or at the micro/nano scale) the Stokes constitutive law becomes invalid/inaccurate. As such, the Stokeslet has limited applicability in the analysis of creeping gas flows at the micro/nano scale. However, constitutive closures exist that extend the applicability of the continuum treatment to higher Knudsen numbers. Notably, Grad’s family of moment equations(and particularly their ‘regularised’ counterparts; e.g. the R13 equations due to Struchtrup and Torrilhon) have attracted significant attention in recent years.

In this talk we introduce fundamental solutions to the the linearised R13 equations for very low Reynolds number flows; equivalent to the Stokeslet, but applicable to higher Knudsen numbers. A simple numerical implementation of the method of fundamental solutions (MFS) is presented for some three-dimensional creeping flows. Incorporation of these new fundamental solutions into the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere, capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated.